Algebraic Geometry is a fascinating branch of Mathematics that combines methods from both Algebra and Geometry. It transcends the limited scope of pure Algebra by means of geometric construction principles. Putting forward this idea, Grothendieck revolutionized Algebraic Geometry in the late 1950s by inventing schemes. Schemes now also play an important role in Algebraic Number Theory, a field that used to be far away from Geometry. The new point of view paved the way for spectacular progress, such as the proof of Fermat's Last Theorem by Wiles and Taylor.
This book explains the scheme-theoretic approach to Algebraic Geometry for non-experts, while more advanced readers can use it to broaden their view on the subject. A separate part presents the necessary prerequisites from Commutative Algebra, thereby providing an accessible and self-contained introduction to advanced Algebraic Geometry.
Every chapter of the book is preceded by a motivating introduction with an informal discussion of its contents and background. Typical examples, and an abundance of exercises illustrate each section. Therefore the book is an excellent companion for self-studying or for complementing skills that have already been acquired. It can just as well serve as a convenient source for (reading) course material and, in any case, as supplementary literature. The present edition is a critical revision of the earlier text.
Author(s): Siegfried Bosch
Series: Universitext
Edition: 2
Publisher: Springer-Verlag London Ltd
Year: 2022
Language: English
Pages: 504
City: London (UK)
Tags: Algebraic Geometry, Commutative Algebra, Hilbert’s Nullstellensatz, Homological Algebra, Noetherian and Artinian Rings, Schemes, Sheaves
Preface
Contents
Part A Commutative Algebra
Introduction
1. Rings and Modules
Background and Overview
1.1 Rings and Ideals
1.2 Local Rings and Localization of Rings
1.3 Radicals
1.4 Modules
1.5 Finiteness Conditions and the Snake Lemma
2. The Theory of Noetherian Rings
Background and Overview
2.1 Primary Decomposition of Ideals
2.2 Artinian Rings and Modules
2.3 The Artin–Rees Lemma
2.4 Krull Dimension
3. Integral Extensions
Background and Overview
3.1 Integral Dependence
3.2 Noether Normalization and Hilbert’s Nullstellensatz
3.3 The Cohen–Seidenberg Theorems
4. Extension of Coefficients and Descent
Background and Overview
4.1 Tensor Products
4.2 Flat Modules
4.3 Extension of Coefficients
4.4 Faithfully Flat Descent of Module Properties
4.5 Categories and Functors
4.6 Faithfully Flat Descent of Modules and of their Morphisms
5. Homological Methods: Ext and Tor
Background and Overview
5.1 Complexes, Homology, and Cohomology
5.2 The Tor Modules
5.3 Injective Resolutions
5.4 The Ext Modules
Part B Algebraic Geometry
Introduction
6. Affine Schemes and Basic Constructions
Background and Overview
6.1 The Spectrum of a Ring
6.2 Functorial Properties of Spectra
6.3 Presheaves and Sheaves
6.4 Inductive and Projective Limits
6.5 Morphisms of Sheaves and Sheafification
6.6 Construction of Affine Schemes
6.7 The Affine n-Space
6.8 Quasi-Coherent Modules
6.9 Direct and Inverse Images of Module Sheaves
7. Techniques of Global Schemes
Background and Overview
7.1 Construction of Schemes by Gluing
7.2 Fiber Products
7.3 Subschemes and Immersions
7.4 Separated Schemes
7.5 Noetherian Schemes and their Dimension
7.6 . Čech Cohomology
7.7 Grothendieck Cohomology
8. Étale and Smooth Morphisms
Background and Overview
8.1 Differential Forms
8.2 Sheaves of Differential Forms
8.3 Morphisms of Finite Type and of Finite Presentation
8.4 Unramified Morphisms
8.5 Smooth Morphisms
9. Projective Schemes and Proper Morphisms
Background and Overview
9.1 Homogeneous Prime Spectra as Schemes
9.2 Invertible Sheaves and Serre Twists
9.3 Divisors
9.4 Global Sections of Invertible Sheaves
9.5 Proper Morphisms
9.6 Abelian Varieties are Projective
Literature
Glossary of Notations
Index
